fisherfit {vegan} | R Documentation |
Function fisherfit
fits Fisher's logseries to abundance
data. Function prestonfit
groups species frequencies into
doubling octave classes and fits Preston's lognormal model, and
function prestondistr
fits the truncated lognormal model
without pooling the data into octaves.
fisherfit(x, ...) ## S3 method for class 'fisherfit': confint(object, parm, level = 0.95, ...) ## S3 method for class 'fisherfit': profile(fitted, alpha = 0.01, maxsteps = 20, del = zmax/5, ...) prestonfit(x, ...) prestondistr(x, truncate = -1, ...) ## S3 method for class 'prestonfit': plot(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue", line.col = "red", lwd = 2, ...) ## S3 method for class 'prestonfit': lines(x, line.col = "red", lwd = 2, ...) veiledspec(x, ...) as.fisher(x, ...)
x |
Community data vector for fitting functions or their result
object for plot functions. |
object, fitted |
Fitted model. |
parm |
Not used. |
level |
The confidence level required. |
alpha |
The extend of profiling as significance. |
maxsteps |
Maximum number of steps in profiling. |
del |
Step length. |
truncate |
Truncation point for log-Normal model, in log2 units. Default value -1 corresponds to the left border of zero Octave. The choice strongly influences the fitting results. |
xlab, ylab |
Labels for x and y axes. |
bar.col |
Colour of data bars. |
line.col |
Colour of fitted line. |
lwd |
Width of fitted line. |
... |
Other parameters passed to functions. |
In Fisher's logarithmic series the expected
number of species f with n observed individuals is
f_n = α x^n / n (Fisher et al. 1943). The estimation
follows Kempton & Taylor (1974) and uses function
nlm
. The estimation is possible only for genuine
counts of individuals. The parameter α is used as a
diversity index, and α and its standard error can be
estimated with a separate function fisher.alpha
. The
parameter x is taken as a nuisance parameter which is not
estimated separately but taken to be n/(n+α). Helper
function as.fisher
transforms abundance data into Fisher
frequency table.
Function fisherfit
estimates the standard error of
alpha. However, the confidence limits cannot be directly
estimated from the standard errors, but you should use function
confint
based on profile likelihood. Function confint
uses function confint.glm
of the MASS
package, using profile.fisherfit
for the profile
likelihood. Function profile.fisherfit
follows
profile.glm
and finds the tau parameter or
signed square root of two times log-Likelihood profile. The profile can
be inspected with a plot
function which shows the tau
and a dotted line corresponding to the Normal assumption: if standard
errors can be directly used in Normal inference these two lines
are similar.
Preston (1948) was not satisfied with Fisher's model which seemed to
imply infinite species richness, and postulated that rare species is a
diminishing class and most species are in the middle of frequency
scale. This was achieved by collapsing higher frequency classes into
wider and wider ``octaves'' of doubling class limits: 1, 2, 3–4,
5–8, 9–16 etc. occurrences. Any logseries data will look like
lognormal when plotted this way. The expected frequency f at abundance
octave o is defined by f = S0 exp(-(log2(o)-mu)^2/2/sigma^2), where
μ is the location of the mode and σ the width, both
in log2 scale, and S0 is the expected number
of species at mode. The lognormal model is usually truncated on the
left so that some rare species are not observed. Function
prestonfit
fits the truncated lognormal model as a second
degree log-polynomial to the octave pooled data using Poisson
error. Function prestondistr
fits left-truncated Normal distribution to
log2 transformed non-pooled observations with direct
maximization of log-likelihood. Function prestondistr
is
modelled after function fitdistr
which can be used
for alternative distribution models. The functions have common print
,
plot
and lines
methods. The lines
function adds
the fitted curve to the octave range with line segments showing the
location of the mode and the width (sd) of the response.
The total
extrapolated richness from a fitted Preston model can be found with
function veiledspec
. The function accepts results both from
prestonfit
and from prestondistr
. If veiledspec
is
called with a species count vector, it will internally use
prestonfit
. Function specpool
provides
alternative ways of estimating the number of unseen species. In fact,
Preston's lognormal model seems to be truncated at both ends, and this
may be the main reason why its result differ from lognormal models
fitted in Rank–Abundance diagrams with functions
rad.lognormal
.
The function prestonfit
returns an object with fitted
coefficients
, and with observed (freq
) and fitted
(fitted
) frequencies, and a string describing the fitting
method
. Function prestondistr
omits the entry fitted
.
The function fisherfit
returns the result of nlm
, where item
estimate
is α. The result object is amended with the
following items:
df.residuals |
Residual degrees of freedom. |
nuisance |
Parameter x. |
fisher |
Observed data from as.fisher . |
It seems that Preston regarded frequencies 1, 2, 4, etc.. as ``tied'' between octaves. This means that only half of the species with frequency 1 were shown in the lowest octave, and the rest were transferred to the second octave. Half of the species from the second octave were transferred to the higher one as well, but this is usually not as large number of species. This practise makes data look more lognormal by reducing the usually high lowest octaves, but is too unfair to be followed. Therefore the octaves used in this function include the upper limit. If you do not accept this, you must change the function yourself. Williamson & Gaston (2005) discuss alternative definitions in detail, and they should be consulted for a critical review of log-Normal model.
Bob O'Hara bob.ohara@helsinki.fi (fisherfit
) and Jari Oksanen.
Fisher, R.A., Corbet, A.S. & Williams, C.B. (1943). The relation between the number of species and the number of individuals in a random sample of animal population. Journal of Animal Ecology 12: 42–58.
Kempton, R.A. & Taylor, L.R. (1974). Log-series and log-normal parameters as diversity discriminators for Lepidoptera. Journal of Animal Ecology 43: 381–399.
Preston, F.W. (1948) The commonness and rarity of species. Ecology 29, 254–283.
Williamson, M. & Gaston, K.J. (2005). The lognormal distribution is not an appropriate null hypothesis for the species–abundance distribution. Journal of Animal Ecology 74, 409–422.
diversity
, fisher.alpha
,
radfit
, specpool
. Function
fitdistr
of MASS package was used as the
model for prestondistr
. Function density
can be used for
smoothed ``non-parametric'' estimation of responses, and
qqplot
is an alternative, traditional and more effective
way of studying concordance of observed abundances to any distribution model.
data(BCI) mod <- fisherfit(BCI[5,]) mod plot(profile(mod)) confint(mod) # prestonfit seems to need large samples mod.oct <- prestonfit(colSums(BCI)) mod.ll <- prestondistr(colSums(BCI)) mod.oct mod.ll plot(mod.oct) lines(mod.ll, line.col="blue3") # Different ## Smoothed density den <- density(log2(colSums(BCI))) lines(den$x, ncol(BCI)*den$y, lwd=2) # Fairly similar to mod.oct ## Extrapolated richness veiledspec(mod.oct) veiledspec(mod.ll)